Looks good!
That's the right shift pattern, except I believe you are supposed to pull up on the knob to select reverse. Clutch hydraulics are the most common cause of shifting problems, tho of course there are many possibilities.
For towing, my only choices would be either flatbed or trailer. If you do opt for a tow dolly, you'll want to disconnect the driveshaft if you are going more than perhaps 30-40 miles (the gearbox will not lubricate properly while the input shaft is not turning); and also make sure the rear tires are up to date. Old radial tires can fly apart suddenly, even if they have lots of tread and no weather-checks; the resulting damage can be severe. Last time it happened to me, the belts turned into a giant flap wheel and took off all the paint plus part of the sheet metal before I could get pulled over. Using a tow bar is a really bad idea IMO, the bumper and frame extensions are not strong enough. It probably won't break completely, but the flexing won't be good.
Does the "start button" have two big cables attached to it? If so, that is the original starter solenoid and you'll probably want to keep it. It does come in handy from time to time
For manuals, I definitely prefer the factory manuals. Fortunately reprints are readily available. Check eBay for the CD version, around $10-$12 with shipping. Moss, Roadster Factory and maybe VB will have a paper reprint that includes the workshop manual and owner's manual, but they are rather more expensive. I just checked TRF and they have it "on sale" for $80 + S&H (PN RBTR4).
Every once in awhile, tatty originals show up on eBay without "silly money" price tags on them.
They won't really help you with troubleshooting much, but will have all the service procedures. These old cars require a lot more periodic maintenance than newer cars do; and many of the common problems can be traced to neglected maintenance. One of my favorites, just to illustrate how arcane some of it can be, is to put a drop of motor oil under the ignition rotor every 6000 miles. (It slowly soaks down inside the distributor and lubricates the centrifugal advance mechanism.)
Lots of debate over the best fluids to use, not really any clear consensus. Personally, I prefer
Valvoline VR1 synthetic for the engine (which has lots of the magic ZDDP);
Redline MT-90 for the gearbox (which is a full synthetic GL4 specifically designed to improve synchro operation in manual gearboxes); and
Valvoline Synpower gear oil (full synthetic, GL5, 75W90; has none of the sulfur additive that can attack the thrust washers in a TR diff).
Oh, and I've been using Prestone "any color" extended life antifreeze.
My 59 TR3A didn't seem to care about unleaded fuel, but the 56 was clearly suffering from valve seat recession until I started adding an ounce of Redline lead substitute to every tankful. Based on some comments from a local expert about excessive ring wear with "oxygenated" gasoline (which is all you can buy around here), I also add about 3 ounces per tankful of TC-W3 2-cycle oil to the gasoline. Can't say for sure that it helps, but the engine seems to like it. And the TC-W3 is a lot cheaper than Marvel Mystery Oil (which is actually what he reommended).
BTW, Triumph motorcars have essentially nothing in common with Triumph motorcycles except the name and being British. The two companies split way back in 1934 (I think it was).